Boundedness of the extremal solution of some pp-Laplacian problems

In this article we consider the pp-Laplace equation −Δpu=λf(u) on a smooth bounded domain of RNRN with zero Dirichlet boundary conditions. Under adequate assumptions on ff we prove that the extremal solution of this problem is in the energy class W01,p(Ω) independently of the domain. We also obtain LqLq and W1,qW1,q estimates for such a solution. Moreover, we prove its boundedness for some range of dimensions depending on the nonlinearity ff.